9-6
The Quadratic Formula
and the Discriminant
Vocabulary
Review
1. Writing What are some advantages to using a formula to express something like
the area of a rectangle?
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Vocabulary Builder
discriminant
discriminant (noun) dih skrim uh nunt
b2 – 4ac
Main Idea: The value of the discriminant tells how many real-number solutions a
quadratic equation has.
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Word Origin: The word discriminant comes from the Latin “discriminare,” which
means “to distinguish.”
Discriminant b2 2 4ac
Number of Real-Number Solutions
positive
zero
negative
2
1
0
Use Your Vocabulary
Find the value of the discriminant of each equation. Then determine the number
of real-number solutions the equation has.
2. 5x2 1 3x 2 9 5 0
Chapter 9
3. 3x2 2 2x 1 7 5 0
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solution(s)
4. x2 2 4x 2 6 5 0
solution(s)
278 solution(s)
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Definition: The discriminant of the equation ax2 1 bx 1 c 5 0
is the value of the related expression b2 2 4ac.
Key Concept Quadratic Formula
If ax2 1 bx 1 c 5 0, and a 2 0, then
x5
2b 4 "b2 2 4ac
2a
You can use the quadratic formula to solve any quadratic equation.
5. Suppose 5x 2 2 8x 1 3 5 0.
Write the values for a, b, and c
in the quadratic formula.
) 4 ÅQ R 2 2 4 ? 2(
x 5
?
2?
Problem 1 Using the Quadratic Formula
Got It? What are the solutions of x2 2 4x 5 21? Use the quadratic formula.
6. Write the equation x2 2 4x 5 21 in standard form and identify a, b, and c.
a 5 50
b 5 c 5 7. Substitute the values from Exercise 6 into the quadratic formula.
x 5
x 5
2b 4 "b2 2 4ac
2a
Use the quadratic formula.
2(24) 4 ÅQ R 2 24 ? ?
Substitute for a, b, and c.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
2?
x 5
4 4Å
x 5
4 4Å
1
Simplify the radicand.
41
x 5
x 5 Simplify the power and the products.
42
or
x5
or
x 5 Write as two equations.
Simplify.
Problem 2 Finding Approximate Solutions
Got It? A batter strikes a baseball. The equation y 5 20.005x 2 1 0.7x 1 3.5
models its path, where x is the horizontal distance, in feet, the ball travels and y is
the height, in feet, of the ball. How far from the batter will the ball land? Round to
the nearest tenth of a foot.
279
Lesson 9-6
8. The quadratic formula is used to solve for x below. Give a justification for each step.
x 5
x 5
x 5
20.7 4 "(0.7)2 2 4 ? (20.005) ? (3.5)
2 ? (20.005)
20.7 4 "0.56
20.01
20.7 1 "0.56
20.01
x < 24.8
20.7 2 "0.56
20.01
or
x5
or x < 144.8
9. The baseball will land about feet from the batter.
There are many methods for solving a quadratic equation. You can always use the
quadratic formula, but sometimes another method may be easier.
When to Use
Graphing
Use if you have a graphing calculator handy.
Square Roots
Use if the equation has no x-term.
Factoring
Use if you can factor the equation easily.
Completing the Square
Use if the x2 coefficient is 1, but you cannot easily factor the equation.
Quadratic Formula
Use if the equation cannot be factored easily or at all.
Problem 3 Choosing an Appropriate Method
Got It? Which method(s) would you choose to solve each equation? Justify
your reasoning.
x2 2 8x 1 12 5 0
169x2 5 36
5x2 1 13x 2 1 5 0
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Draw a line from each equation in Column A to the most appropriate method for
solving the equation in Column B.
Column A
Column B
10. x2 2 8x 1 12 5 0
square roots; there is no x-term
11. 169x2 5 36
factoring; the equation is easily factorable
12. 5x2 1 13x 2 1 5 0
quadratic formula, graphing; the equation cannot be factored
Problem 4 Using the Discriminant
Got It? How many real-number solutions does 6x2 2 5x 5 7 have?
13. Write the equation in standard form.
Chapter 9
280 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Method
14. Evaluate the discriminant.
b 2 2 4ac 5 ( )2 2 4 ? 5 5 ? 1 15. Circle the number of real-number solutions 6x 2 2 5x 5 7 has.
no solutions
1 solution
2 solutions
7 solutions Lesson Check • Do you UNDERSTAND?
Reasoning What method would you use to solve the equation x2 1 9x 1 c 5 0
if c 5 14? If c 5 7? Explain.
16. Write the equation with c 5 14.
17. Circle the method you would use to solve the equation.
graphing
square roots
factoring
completing the square
quadratic formula 18. Explain your choice in Exercise 17.
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Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
19. Write the equation with c 5 7.
20. Circle the method you would use to solve the equation.
graphing
square roots
factoring
completing the square
quadratic formula 21. Explain your choice in Exercise 20.
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Math Success
Check off the vocabulary words that you understand.
quadratic formula
discriminant
Rate how well you can use the quadratic formula.
Need to
review
0
2
4
6
8
Now I
get it!
10
281
Lesson 9-6